Valley Nernst effect is a newly proposed and experimentally confirmed effect, which could be used to design novel thermoelectric devices. We study the valley Nernst effect in (M+N)-layer twisted multilayer graphene systems by a simple low-energy effective model. It is found that the total valley Nernst coefficient (VNC) is three orders of magnitude larger than that in monolayer group-VI dichalcogenides. The total VNC increases with the increase of layer numbers. It is shown that the total VNC exhibits a structure with three peaks as a function of the Fermi energy. We identify that the central peak is always negative stemming from the flat band. Two shoulder peaks are positively induced by the conduction and valence bands, respectively. These predicted features can be tested experimentally. The present work would shed more light on valley caloritronics.

Valleytronics is an emerging field of research which utilizes the valley degree of freedom to encode information. However, it is technically nontrivial to produce a stable valley polarization and to achieve efficient control and manipulation of valleys. Spin-valley locking refers to the coupling between spin and valley degrees of freedom in the materials with large spin-orbit coupling (SOC) and enables the manipulation of valleys indirectly through controlling spins. Here, we review the recent advances in spin-valley locking physics and outline possible device implications. In particular, we focus on the spin-valley locking induced by SOC and external electric field in certain two-dimensional materials with inversion symmetry and demonstrate the intriguing switchable valley-spin polarization, which can be utilized to design the promising electronic devices, namely, valley-spin valves and logic gates.

Monolayer group-VIB transition metal dichalcogenides (TMDs) feature low-energy massive Dirac fermions, which have valley contrasting Berry curvature. This nontrivial local band topology gives rise to valley Hall transport and optical selection rules for interband transitions that open up new possibilities for valleytronics. However, the large bandgap in TMDs results in relatively small Berry curvature, leading to weak valley contrasting physics in practical experiments. Here, we show that Dirac fermions with tunable large Berry curvature can be engineered in moiré superlattice of TMD heterobilayers. These moiré Dirac fermions are created in a magnified honeycomb lattice with its sublattice degree of freedom formed by two local moiré potential minima. We show that applying an on-site potential can tune the moiré flat bands into helical ones. In short-period moiré superlattice, we find that the two moiré valleys become asymmetric, which results in a net spin Hall current. More interestingly, a circularly polarized light drives these moiré Dirac fermions into quantum anomalous Hall phase with chiral edge states. Our results open a new possibility to design the moiré-scale spin and valley physics using TMD moiré structures.

The Ising spin-orbit coupling could give rise to the spin-triplet Cooper pairs and equal-spin Andreev reflection (AR) in Ising superconductors. Here we theoretically study the valley-dependent equal-spin AR in a ferromagnet/Ising superconductor junction with a circularly polarized light applied to the ferromagnet. Because of the spin-triplet Cooper pairs and the optical irradiation, eight kinds of AR processes appear in the junction, including equal-spin AR and normal AR, the strengths and properties of which strongly depend on the valley degree of freedom. The AR probabilities for the incident electron from the two valleys exhibit certain symmetry with respect to the magnetization angle and the effective energy of light. The equal-spin AR and normal AR present different features and resonant behaviors near the superconducting gap edges. Due to equal-spin-triplet Cooper pairs, not only charge supercurrent but also spin supercurrent can transport in the Ising superconductors. The differential spin conductance for electron injecting from the two valleys can be controlled by the circularly polarized light.

Spin-valley polarization and bandgap regulation are critical in the developing of quantum devices. Here, by employing the density functional theory, we investigate the effects of stacking form, thickness and magnetic moment in the electronic structures of WSe$_{2}$-MoS$_{2}$ heterostructures. Calculations show that spin-valley polarization maintains in all situations. Increasing thickness of 2H-MoS$_{2}$ not only tunes the bandgap but also changes the degeneracy of the conduction band minimums (CBM) at $K/K_1$ points. Gradual increase of micro magnetic moment tunes the bandgap and raises the valence band maximums (VBM) at $\varGamma$ point. In addition, the regulation of band gap by the thickness of 2H-MoS$_{2}$ and introduced magnetic moment depends on the stacking type. Results suggest that WSe$_{2}$-MoS$_{2}$ heterostructure supports an ideal platform for valleytronics applications. Our methods also give new ways of optical absorption regulation in spin-valley devices.

In recent years, valleytronics researches based on 2D semiconducting transition metal dichalcogenides have attracted considerable attention. On the one hand, strong spin-orbit interaction allows the presence of spin-valley coupling in this system, which provides spin addressable valley degrees of freedom for information storage and processing. On the other hand, large exciton binding energy up to hundreds of meV enables excitons to be stable carriers of valley information. Valley polarization, marked by an imbalanced exciton population in two inequivalent valleys (+$K$ and -$K$), is the core of valleytronics as it can be utilized to store binary information. Motivated by the potential applications, we present a thorough overview of the recent advancements in the generation, relaxation, manipulation, and transport of the valley polarization in nonmagnetic transition metal dichalcogenide layered semiconductors. We also discuss the development of valleytronic devices and future challenges in this field.

We study the valley-dependent electron transport in a four-terminal mesoscopic device of the two monolayer graphene nanoribbons vertically stacked together, where the intersection forms a bilayer graphene lattice with a controllable twist angle. Using a tight-binding lattice model, we show that the longitudinal and transverse conductances exhibit significant valley polarization in the low energy regime for small twist angles. As the twist angle increases, the valley polarization shifts to the high energy regime. This arises from the regrouping effect of the electron band in the twisted bilayer graphene region. But for relatively large twist angles, no significant valley polarization is observed. These results are consistent with the spectral densities of the twisted bilayer graphene.

The presence of two sublattices in hexagonal graphene brings two energetically degenerate extremes in the conduction and valence bands, which are identified by the valley quantum number. Recently, this valley degree of freedom has been suggested to encode and process information, which develops a new carbon-based electronics named graphene valleytronics. In this topical review, we present and discuss valley-related transport properties in bulk graphene monolayers, which are due to strain-induced pseudomagnetic fields and associated vector potential, sublattice-stagger potential, and the valley-Zeeman effect. These valley-related interactions can be utilized to obtain valley filtering, valley spatial separation, valley-resolved guiding modes, and valley-polarized collective modes such as edge or surface plasmons. The present challenges and the perspectives on graphene valleytronics are also provided in this review.

As opposed to the prototypical MoS₂ with centroasymmetry, Janus ferrovalley materials such as H-VSSe are less symmetric with the mirror symmetry and time reversal symmetry broken, and hence possess spontaneous valley polarization and strong ferroelasticity. The optical transition is an important means to excite the valley carriers. We investigate the optical spectrum of H-VSSe by using the many-body perturbation-based GW approach and solving the Bethe—Salpeter equation (BSE) to include the electron—hole interactions. It is found that after the GW correction, the band gaps of the quasiparticle bands are much larger than those obtained by the normal density functional theory. The system is ferromagnetic and the valley gaps become non-degenerate due to spin—orbit coupling (SOC). The position of the lowest BSE peak is much lower than the quasiparticle band gap, indicating that the excitonic effect is large. The peak is split into two peaks by the SOC. The binding energy difference between these two BSE peaks is about the same as the difference between the inequivalent valley gaps. Our results show that in Janus H-VSSe the two lowest exciton peaks are from the two inequivalent valleys with different gaps, in contrast to the A and B exciton peaks of MoS₂ which are from the same valley.

Valleytronics is an emergent discipline in condensed matter physics and offers a new way to encode and manipulate information based on the valley degree of freedom in materials. Among the various materials being studied, Kekulé distorted graphene has emerged as a promising material for valleytronics applications. Graphene can be artificially distorted to form the Kekulé structures rendering the valley-related interaction. In this work, we review the recent progress of research on Kekulé structures of graphene and focus on the modified electronic bands due to different Kekulé distortions as well as their effects on the transport properties of electrons. We systematically discuss how the valley-related interaction in the Kekulé structures was used to control and affect the valley transport including the valley generation, manipulation, and detection. This article summarizes the current challenges and prospects for further research on Kekulé distorted graphene and its potential applications in valleytronics.

The electron's charge and spin degrees of freedom are at the core of modern electronic devices. With the in-depth investigation of two-dimensional materials, another degree of freedom, valley, has also attracted tremendous research interest. The intrinsic spontaneous valley polarization in two-dimensional magnetic systems, ferrovalley material, provides convenience for detecting and modulating the valley. In this review, we first introduce the development of valleytronics. Then, the valley polarization forms by the p-, d-, and f-orbit that are discussed. Following, we discuss the investigation progress of modulating the valley polarization of two-dimensional ferrovalley materials by multiple physical fields, such as electric, stacking mode, strain, and interface. Finally, we look forward to the future developments of valleytronics.

Valleytronics, using valley degree of freedom to encode, process, and store information, may find practical applications in low-power-consumption devices. Recent theoretical and experimental studies have demonstrated that two-dimensional (2D) honeycomb lattice systems with inversion symmetry breaking, such as transition-metal dichalcogenides (TMDs), are ideal candidates for realizing valley polarization. In addition to the optical field, lifting the valley degeneracy of TMDs by introducing magnetism is an efficient way to manipulate the valley degree of freedom. In this paper, we first review the recent progress on valley polarization in various TMD-based systems, including magnetically doped TMDs, intrinsic TMDs with both inversion and time-reversal symmetry broken, and magnetic TMD heterostructures. When topologically nontrivial bands are empowered into valley-polarized systems, valley-polarized topological states, namely valley-polarized quantum anomalous Hall effect can be realized. Therefore, we have also reviewed the theoretical proposals for realizing valley-polarized topological states in 2D honeycomb lattices. Our paper can help readers quickly grasp the latest research developments in this field.

Inner edge state with spin and valley degrees of freedom is a promising candidate for designing a dissipationless device due to the topological protection. The central challenge for the application of the inner edge state is to generate and modulate the polarized currents. In this work, we discover a new mechanism to generate fully valley- and spin—valley-polarized current caused by the Bloch wavevector mismatch (BWM). Based on this mechanism, we design some serial-typed inner-edge filters. By using once of the BWM, the coincident states could be divided into transmitted and reflected modes, which can serve as a valley or spin—valley filter. In particular, while with twice of the BWM, the incident current is absolutely reflected to support an off state with a specified valley and spin, which is different from the gap effect. These findings give rise to a new platform for designing valleytronics and spin-valleytronics.

The anomalous valley Hall effect (AVHE) can be used to explore and utilize valley degrees of freedom in materials, which has potential applications in fields such as information storage, quantum computing and optoelectronics. AVHE exists in two-dimensional (2D) materials possessing valley polarization (VP), and such 2D materials usually belong to the hexagonal honeycomb lattice. Therefore, it is necessary to achieve valleytronic materials with VP that are more readily to be synthesized and applicated experimentally. In this topical review, we introduce recent developments on realizing VP as well as AVHE through different methods, i.e., doping transition metal atoms, building ferrovalley heterostructures and searching for ferrovalley materials. Moreover, 2D ferrovalley systems under external modulation are also discussed. 2D valleytronic materials with AVHE demonstrate excellent performance and potential applications, which offer the possibility of realizing novel low-energy-consuming devices, facilitating further development of device technology, realizing miniaturization and enhancing functionality of them.

Bismuthene, an inherently hexagonal structure characterized by a huge bulk gap, offers a versatile platform for investigating the electronic transport of various topological quantum states. Using nonequilibrium Green's function method and Landauer—Büttiker formula, we thoroughly investigate the transport properties of various Hall-type quantum states, including quantum spin Hall (QSH) edge states, quantum valley Hall kink (QVHK) states, and quantum spin—valley Hall kink (QSVHK) states, in the presence of various disorders. Based on the exotic transport features, a spin—valley filter, capable of generating a highly spin- and valley-polarized current, is proposed. The valley index and the spin index of the filtered QSVHK state are determined by the staggered potential and the intrinsic spin—orbit coupling, respectively. The efficiency of the spin—valley filter is supported by the spacial current distribution, the valley-resolved conductance, and the spin-resolved conductance. Compared with a sandwich structure for QSVHK, our proposed spin—valley filter can work with a much smaller size and is more accessible in the experiment.

Valley-polarized quantum anomalous Hall effect (VQAHE), combined nontrivial band topology with valleytronics, is of importance for both fundamental sciences and emerging applications. However, the experimental realization of this property is challenging. Here, by using first-principles calculations and modal analysis, we predict a mechanism of producing VQAHE in two-dimensional ferromagnetic van der Waals germanene/MnI$_{2}$ heterostructure. This heterostructure exhibits both valley anomalous Hall effect and VQAHE due to the joint effects of magnetic exchange effect and spin-orbital coupling with the aid of anomalous Hall conductance and chiral edge state. Moreover interestingly, through the electrical modulation of ferroelectric polarization state in In$_{2}$Se$_{3}$, the germanene/MnI$_{2}$/In$_{2}$Se$_{3}$ heterostructure can undergo reversible switching from a semiconductor to a metallic behavior. This work offers a guiding advancement for searching for VQAHE in ferromagnetic van der Waals heterostructures and exploiting energy-efficient devices based on the VQAHE.